had. No more telling example of this could be wished for than the insane delusion under which they permit Euclid to be employed in our elementary teaching. They seem voluntarily to weight alike themselves and their pupils for the race; and a cynic might, perhaps without much injustice, say they do so that they may have mere self-imposed and avoidable difficulties to face instead of the new, real, and dreaded ones (belonging to regions hitherto unpenetrated) with which Quaternions would too soon enable them to come into contact. But this game will certainly end in disaster. As surely as Mathematics came to a relative stand-still in this country for nearly a century after Newton, so surely will it do so again if we leave our eager and watchful rivals abroad to take the initiative in developing the grand method of Hamilton. And it is not alone French and Germans whom we have now to dread, Russia, America, regenerated Italy, and other nations are all fairly entered for the contest.

The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulæ than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting, while even the most abstruse branches of physics, as yet totally incapable of being popularized, attract the attention of the uninitiated. The reason may perhaps be sought in the fact that, while perhaps the only successful attempt to invest mathematical reasoning with a halo of glory-that made in this Section by Prof. Sylvester-is known to a comparative few, several of the highest problems of physics are connected with those simple observations which are possible to the many. The smell of lightning has been observed for thousands of years, it required the sagacity of Schönbein to trace it to the formation of Ozone. Not to speak of the (probably fabulous) apple of Newton, what enormous consequences did he obtain by passing light through a mere wedge of glass, and by simply laying a lens on a flat plate! The patching of a trumpery model led Watt to his magnificent inventions. As children at the sea-shore playing with a "roaring buckie," or in later life lazily puffing out rings of tobacco-smoke, we are illustrating two of the splendid researches of Helmholtz. And our President, by the bold, because simple, use of reaction instead of action, has eclipsed even his former services to the Submarine Telegraph, and given it powers which but a few years ago would have been deemed unattainable.

In experimental Physics our case is not hopeless, perhaps not as yet even alarming. Still something of the same kind may be said in this as in pure Mathematics. If Thomson's Theory of Dissipation, for instance, be not speedily developed in this country, we shall soon learn its consequences from abroad. The grand test of our science, the proof of its being a reality and not a mere inventing of new terms and squabbling as to what they shall mean, is that it is ever advancing. There is no standing still; there is no running round and round as in a beaten donkey-track, coming back at the end of a century or so into the old positions, and fighting the self-same battles under slightly different banners, which is merely another form of stagnation (Kinetic Stability in fact). "A little folding of the hands to sleep," in chuckling satisfaction at what has been achieved of late years by our great experimenters, and we shall be left hopelessly behind. The sad fate of Newton's successors ought ever to be a warning to us. Trusting to what he had done, they allowed mathematical science almost to die out in this country, at least as compared with its immense progress in Germany and France. It required the united exertions of the late Sir J. Herschel and many others to render possible in these islands a Boole and a Hamilton. If the successors of Davy and Faraday pause to ponder even on their achievements, we shall soon be again in the same state of ignominious inferiority. Who will then step in to save us?

Even as it is, though we have among us many names quite as justly great as any that our rivals can produce, we have also (even in our educated classes) such an immense amount of ignorance and consequent credulity, that it seems matter for surprise that true science is able to exist. Spiritualists, Circle-squarers, Perpetualmotionists, Believers that the earth is flat and that the moon has no rotation, swarm about us. They certainly multiply much faster than do genuine men of science. This is characteristic of all inferior races, but it is consolatory to remember that in spite of it these soon become extinct. Your quack has his little

day, and disappears except to the antiquary. But in science nothing of value can ever be lost; it is certain to become a stepping-stone on the way to further truth. Still, when our stepping-stones are laid, we should not wait till others employ them. "Gentlemen of the Guard be kind enough to fire first" is a courtesy entirely out of date; with the weapons of the present day it would be simply suicide.

There is another point which should not be omitted in an address like this. For obvious reasons I must speak of the general question only, not venturing on examples, though I could give many telling ones. Even among our greatest men of science in this country there is comparatively little knowledge of what has been already achieved, except of course in the one or more special departments cultivated by each individual. There can be little doubt that one cause at least of this is to be sought in the extremely meagre interest which our statesmen, as a rule, take in scientific progress. While abroad we find half a dozen professors teaching parts of the same subject in one University (each having therefore reasonable leisure), with us one man has to do the whole, and to endeavour as he best can to make something out of his very few spare moments. Along with this, and in great part due to it, there is often found a proneness to believe that what seems evident to the thinker cannot but have been long known to others. Thus the credit of many valuable discoveries is lost to Britain because her philosophers, having no time to spare, do not know that they are discoveries. The scientific men of other nations are, as a rule, better informed [certainly far better encouraged and less over-worked], and perhaps likewise are not so much given to self-depreciation. Until something resembling the 'Fortschritte der Physik,' but in an improved form, and published at smaller intervals and with much less delay, is established in this country, there is little hope of improvement in this respect. Why should science be imperfectly summarized in little haphazard scraps here and there, when mere property has its elaborate series of Money-articles and exact Broker's Share-lists? Such a work would be very easy of accomplishment: we have only to begin boldly; we do not need to go back, for in every year good work is being done at almost every part of the boundary between, as it were, the cultivated land and the still unpenetrated forest-enough at all events to show with all necessary accuracy whereabouts that boundary lies.

There is no need of entering here on the question of Conservation of Energy; it is thoroughly accepted by scientific men, and has revolutionized the greater part of Physics. The facts as to its history also are generally agreed upon, but differences of a formidable kind exist as to the deductions to be drawn from them. These are matters, however, which will be more easily disposed of thirty years hence than now. The Transformation of Energy is also generally accepted, and, in fact, under various unsatisfactory names was almost popularly known before the Conservation of Energy was known in its entirety to more than a very few. But the Dissipation of Energy is by no means well known, and many of the results of its legitimate application have been received with doubt, sometimes even with attempted ridicule. Yet it appears to be at the present moment by far the most promising and fertile portion of Natural Philosophy, having obvious applications of which as yet only a small percentage appear to have been made. Some, indeed, were made before the enunciation of the Principle, and have since been recognized as instances of it. Of such we have good examples in Fourier's great work on Heat-conduction, in the optical theorem that an image can never be brighter than the object, in Gauss's mode of investigating electrical distribution, and in some of Thomson's theorems as to the energy of an electromagnetic field. But its discoverer has, so far as I know, as yet confined himself in its explicit application to questions of Heat-conduction and Restoration of Energy, Geological Time, the Earth's Rotation, and such like. Unfortunately his long-expected Rede Lecture has not yet been published, and its contents (save to those who were fortunate enough to hear it) are still almost entirely unknown.

But there can be little question that the Principle contains implicitly the whole theory of Thermo-electricity, of Chemical Combination, of Allotropy, of Fluorescence, &c., and perhaps even of matters of a higher order than common physics and chemistry. In Astronomy it leads us to the grand question of the age, or

perhaps more correctly the phase of life, of a star or nebula, shows us the material of potential suns, other suns in the process of formation, in vigorous youth, and in every stage of slowly protracted decay. It leads us to look on each planet and satellite as having been at one time a tiny sun, a member of some binary or multiple group, and even now (when almost deprived, at least at its surface, of its original energy) presenting an endless variety of subjects for the application of its methods. It leads us forward in thought to the far-distant time when the materials of the present stellar system shall have lost all but their mutual potential energy, but shall in virtue of it form the materials of future larger suns with their attendant planets. Finally, as it alone is able to lead us, by sure steps of deductive reasoning, to the necessary future of the universe-necessary, that is, if physical laws for ever remain unchanged-so it enables us distinctly to say that the present order of things has not been evolved through infinite past time by the agency of laws now at work, but must have had a distinctive beginning, a state beyond which we are totally unable to penetrate, a state, in fact, which must have been produced by other than the now acting causes.

Thus also, it is possible that in Physiology it may, ere long, lead to results of a different and much higher order of novelty and interest than those yet obtained, immensely valuable though they certainly are.

It was a grand step in science which showed that just as the consumption of fuel is necessary to the working of a steam-engine, or to the steady light of a candle, so the living engine requires food to supply its expenditure in the forms of muscular work and animal heat. Still grander was Rumford's early anticipation that the animal is a more economic engine than any lifeless one we can construct. Even in the explanation of this there is involved a question of very great interest, still unsolved, though Joule and many other philosophers of the highest order have worked at it. Joule has given a suggestion of great value, viz. that the animal resembles an electromagnetic- rather than a heat-engine; but this throws us back again upon our difficulties as to the nature of electricity. Still, even supposing this question fully answered, there remains another-perhaps the highest which the human intellect is capable of directly attacking, for it is simply preposterous to suppose that we shall ever be able to understand scientifically the source of Consciousness and Volition, not to speak of loftier things-there remains the question of Life. Now it may be startling to some of you, especially if you have not particularly considered the matter, to hear it surmised that possibly we may, by the help of physical principles, especially that of the Dissipation of Energy, some time attain to a notion of what constitutes Life-mere Vitality I repeat, nothing higher. If you think for a moment of the vitality of a plant or a zoophyte, the remark perhaps will not appear so strange after all. But do not fancy that the Dissipation of Energy to which I refer is at all that of a watch or such-like piece of mere human mechanism, dissipating the low and common form of energy of a single coiled spring. It must be such that every little part of the living organism has its own store of energy constantly being dissipated, and as constantly replenished from external sources drawn upon by the whole arrangement in their harmonious working together. As an illustration of my meaning, though an extremely inadequate one, suppose Vaucanson's Duck to have been made up of excessively small parts, each microscopically constructed as perfectly as was the comparatively coarse whole, we should have had something barely distinguishable, save by want of instincts, from the living model. But let no one imagine that, should we ever penetrate this mystery, we shall thereby be enabled to produce, except from life, even the lowest form of life. Our President's splendid suggestion of Vortex-atoms, if it be correct, will enable us thoroughly to understand matter, and mathematically to investigate all its properties. Yet its very basis implies the absolute necessity of an intervention of Creative Power to form or to destroy one atom even of dead matter. The question really stands thus:-Is Life physical or no? For if it be in any sense, however slight or restricted, physical, it is to that extent a subject for the Natural Philosopher, and for him alone. It would be entirely out of place for me to discuss such a question as this now and here; I have introduced it merely that I may say a word or two about what has been so often and so persistently croaked against the British Association, viz. that it

tends to develope what are called Scientific Heresies. No doubt such charges are brought more usually against other Sections than against this; but Section A has not been held blameless. It seems to me that the proper answer to all such charges will be very simply and easily given, if we merely show that in our reasonings from observation and experiment we invariably confine our physical conclusions strictly to matter and energy (things which we can weigh and measure) in their multiform combinations. Excepting that which is obviously purely mathematical, whatever is certainly neither matter nor energy, nor dependent upon these, is not a subject to be discussed here, even by implication. All our reasonings in Physics must, so far as we know, be based upon the assumption, founded on experience, that in the universe, whatever be the epoch or the locality, under exactly similar circumstances exactly similar results will be obtained. If this be not granted there is an end of Physical Science, or, rather, there never could have been such a Science*. To use the word "Heresy" with reference to purely physical reasonings about Geological Time, or matters of that kind, is nowadays a piece of folly which even Galileo's judges, were they alive, would shrink from, as calculated to damage none but themselves and the cause which of old they, according to their lights, very naturally maintained.

There must always be wide limits of uncertainty (unless we choose to look upon Physics as a necessarily finite Science) concerning the exact boundary between the Attainable and the Unattainable. One herd of ignorant people, with the sole prestige of rapidly increasing numbers, and with the adhesion of a few fanatical deserters from the ranks of Science, refuse to admit that all the phenomena even of ordinary dead matter are strictly and exclusively in the domain of physical science. On the other hand, there is a numerous group, not in the slightest degree entitled to rank as Physicists (though in general they assume the proud title of Philosophers), who assert that not merely Life, but even Volition and Consciousness are mere physical manifestations. These opposite errors, into neither of which is it possible for a genuine scientific man to fall, so long at least as he retains his reason, are easily seen to be very closely allied. They are both to be attributed to that Credulity which is characteristic alike of Ignorance and of Incapacity. Unfortunately there is no cure; the case is hopeless, for great ignorance almost necessarily presumes incapacity, whether it show itself in the comparatively harmless folly of the Spiritualist or in the pernicious nonsense of the Materialist.

Alike condemned and contemned, we leave them to their proper fate-oblivion; but still we have to face the question, where to draw the line between that which is physical and that which is utterly beyond physics. And, again, our answer is— Experience alone can tell us; for experience is our only possible guide. If we attend earnestly and honestly to its teachings, we shall never go far astray. Man has been left to the resources of his intellect for the discovery not merely of physical laws, but of how far he is capable of comprehending them. And our answer to those who denounce our legitimate studies as heretical is simply this,-A revelation of any thing which we can discover for ourselves, by studying the ordinary course of nature, would be an absurdity.

A profound lesson may be learned from one of the earliest little papers of President, published while he was an undergraduate at Cambridge, where he shows that Fourier's magnificent treatment of the Conduction of Heat leads to formula for its distribution which are intelligible (and of course capable of being fully verified by experiment) for all time future, but which, except in particular cases, when extended to time past, remain intelligible for a finite period only, and then

* It might be possible, and, if so, perhaps interesting, to speculate on the results of secular changes in physical laws, or in particles of matter which are subject to them, but (so far as experience, which is our only guide, has taught us since the beginning of modern science) there seems no trace of such. Even if there were, as these changes must be of necessity extremely slow (because not yet even suspected), we may reasonably expect, from the analogy of the history of such a question as gravitation, especially in the discovery of Neptune, that our work, far from becoming impossible, will merely become considerably more difficult as well as more laborious, but, on that account, all the more creditable when successfully carried out.

indicate a state of things which could not have resulted under known laws from any conceivable previous distribution. So far as heat is concerned, modern investigations have shown that a previous distribution of the matter involved may, by its potential energy, be capable of producing such a state of things at the moment of its aggregation; but the example is now adduced not for its bearing on heat alone, but as a simple illustration of the fact that all portions of our Science, and especially that beautiful one the Dissipation of Energy, point unanimously to a beginning, to a state of things incapable of being derived by present laws from any conceivable previous arrangement.

I conclude by quoting some noble words used by Stokes in his Address at Exeter, words which should be stereotyped for every Meeting of this Association:"When from the phenomena of life we pass on to those of mind, we enter a region "still more profoundly mysterious. . . . Science can be expected to do but little "to aid us here, since the instrument of research is itself the object of investigation. "It can but enlighten us as to the depth of our ignorance, and lead us to look to a "higher aid for that which most nearly concerns our wellbeing."

MATHEMATICS.

Exhibition and Description of a Model of a Conoidal Cubic Surface called the "Cylindroid," which is presented in the Theory of the Geometrical Freedom of a Rigid Body. By ROBERT STAWELL BALL, A.M., Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland.

We become acquainted with the geometrical freedom which a rigid body enjoys by ascertaining the character of all the displacements which the nature of the restraints will permit the body to accept.

If a displacement be infinitely small, it is produced by screwing the body along a certain screw.

If a displacement have finite magnitude, it is produced by an infinite series of infinitely small screw displacements.

For the analysis of geometrical freedom, we shall only consider infinitely small screw displacements. This includes the initial stages of all displacements.

To analyze the geometrical restraints of a rigid body we proceed as follows:Take any line in space. Conceive this line to be the axis about which screws are successively formed of every pitch from ∞ to +. (The pitch of a screw is the distance its nut advances when turned through the angular unit.) We endeavour successively to displace the body about each of these screws, and record the particular screw or screws, if any, about which the restraints have permitted the body to receive a displacement. The same process is to be repeated for every other line in space. If it be found that the restraints have not permitted the body to receive any one of these displacements, then the body is rigidly fixed in space.

If, after all the screws have been tried, the body be found capable of displacement about one screw only, the body possesses the lowest degree of freedom.

If one screw (A) be discovered, and, the trials being continued, a second screw (B) be found, the remaining trials may be abridged by considering the information which the discovery of two screws affords. It is in connexion with the two screws that the cylindroid is presented.

The body may receive any displacement about one or both of the two screws A and B.

The composition of these displacements gives a resultant which could have been produced by displacement about a single screw.

The locus of this single screw is the conoidal cubic surface which has been called the "cylindroid" (at the suggestion of Professor Cayley).

The equation of the cylindroid is

z(x2+y2) −2axy=0.